I believe that the following problem have already been considered by some sophisticated topologist.
Definition 1. A non-compact Hausdorff topological space $X$ is called almost compact if its Stone-Cech compactification coincides with its one point compactification.
An example of almost compact space is $[0,\omega_1)$ for first uncountable ordinal $\omega_1$.
Definition 2. A compact Hausdorff space $X$ is called pretty compact if $X\setminus\{p\}$ is almost compact for all non-isolated points $p\in X$.
I would like to hear answers to any of the following questions.
Questions:
- What are examples of pretty compact spaces?
- Is it true that pretty compact spaces are extremally disconnected?
- Is it true that pretty compact spaces contain dense extremally disconnected subspace?
- Does there exist any characterization of pretty compact spaces?
The question was partially answered on mathoverflow.net. It seems like there is no short clear description for such spaces since they include essentially different classes of topological spaces.